#509311

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,8)(9,11)(10,12)(13,14,16,19)(15,24,25,37)(17,20,28,34)(18,32,29,23)(21,26,35,43)(22,39,36,30)(27,40,33,42)(31,38,44,41);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,25)(20,34)(22,36)(23,32)(26,43)(27,33)(31,44);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(17,30)(18,29)(19,34)(21,37)(22,23)(24,35)(26,31)(27,33)(28,39)(32,36)(38,42)(40,41)(43,44);(6,11)(7,12)(13,15,16,25)(14,21,19,35)(17,31,28,44)(18,33,29,27)(20,37,34,24)(22,40,36,42)(23,41,32,38)(26,39,43,30)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416507);